To create a test case generator for the problem statement provided, we need to ensure that the generated inputs adhere to the constraints and generate valid sequences. Here's a Python function to generate randomized test cases:

```python
import random

def tcgen():
    # Constraints
    MIN_X_D = -10**8
    MAX_X_D = 10**8
    MIN_N = 1
    MAX_N = 2 * 10**5
    
    # Generate random values within the constraints
    N = random.randint(MIN_N, MAX_N)
    X = random.randint(MIN_X_D, MAX_X_D)
    D = random.randint(MIN_X_D, MAX_X_D)
    
    return N, X, D

# Example usage:
for _ in range(5):  # Generate 5 test cases
    print(tcgen())
```

This function generates random values for \( N \), \( X \), and \( D \) within the specified constraints. You can call this function multiple times to generate different test cases. The results will be tuples of three values representing \( N \), \( X \), and \( D \), which can be directly used as test cases for the problem.

Here's how it works:
- \( N \) is randomly generated between 1 and 200,000.
- \( X \) and \( D \) are randomly generated between -100,000,000 and 100,000,000.

### Example Output

Running the function multiple times might give output like this:

```
(193045, 58474524, 2148231)
(36521, -98127463, 47281945)
(146789, 567234, -98765432)
(100000, -100000000, 100000000)
(200000, -12345678, 87654321)
```

These outputs can be used directly as inputs to the problem statement.